Consumer Optimisation Microeconomia III (Lecture 6) Tratto da Cowell F. (2004), Principles of Microeoconomics

What we're going to do: We want to solve the consumer's optimisation problem... We want to solve the consumer's optimisation problem......using methods that we've already introduced....using methods that we've already introduced. This enables us to re-cycle old techniques and results: This enables us to re-cycle old techniques and results:  Run the presentation for firm optimisation…  look for the points of comparison...  and try to find as many reinterpretations as possible. Jump to “Firm”

The problem Maximise consumer’s utility U(x)U(x) U assumed to satisfy the standard “shape” axioms Subject to feasibility constraint x Xx X and to the budget constraint n  p i x i ≤ y i=1 Assume consumption set is the non-negative orthant. The version with fixed money income

Overview... Primal and Dual problems Lessons from the Firm Primal and Dual again Consumer: Optimisation Two fundamental views of consumer optimisation

An obvious approach? We now have the elements of a standard constrained optimisation problem: We now have the elements of a standard constrained optimisation problem:  the constraints on the consumer.  the objective function. The next steps might seem obvious: The next steps might seem obvious:  set up a standard Lagrangean.  solve it.  interpret the solution. But the obvious approach is not always the most insightful. But the obvious approach is not always the most insightful. We’re going to try something a little sneakier… We’re going to try something a little sneakier…

Think laterally... In microeconomics an optimisation problem can often be represented in more than one form. In microeconomics an optimisation problem can often be represented in more than one form. Which form you use depends on the information you want to get from the solution. Which form you use depends on the information you want to get from the solution. This applies here. This applies here. The same consumer optimisation problem can be seen in two different ways. The same consumer optimisation problem can be seen in two different ways. I’ve used the labels “primal” and “dual” that have become standard in the literature. I’ve used the labels “primal” and “dual” that have become standard in the literature.

A five-point plan Set out the basic consumer optimisation problem. Set out the basic consumer optimisation problem. Show that the solution is equivalent to another problem. Show that the solution is equivalent to another problem. Show that this equivalent problem is identical to that of the firm. Show that this equivalent problem is identical to that of the firm. Write down the solution. Write down the solution. Go back to the problem we first thought of... Go back to the problem we first thought of... The primal problem The primal problem again The dual problem

The primal problem x1x1 x2x2 l l x*   But there's another way at looking at this   The consumer aims to maximise utility...   Subject to budget constraint max U(x) subject to n  p i x i  y i=1   Defines the primal problem.   Solution to primal problem Constraint set increasing preference increasing preference Contours of objective function

x1x1 x2x2 The dual problem x* ll ll   Alternatively the consumer could aim to minimise cost...   Subject to utility constraint   Defines the dual problem.   Solution to the problem minimise n  pixi pixi i=1 subject to U(x)    Constraint set Reducing cost Contours of objective function   Cost minimisation by the firm z1z1 z2z2 z* Reducing cost ll ll q   But where have we seen the dual problem before?

Two types of cost minimisation The similarity between the two problems is not just a curiosity. The similarity between the two problems is not just a curiosity. We can use it to save ourselves work. We can use it to save ourselves work. All the results that we had for the firm's “stage 1” problem can be used. All the results that we had for the firm's “stage 1” problem can be used. We just need to translate them intelligently We just need to translate them intelligently

Overview... Primal and Dual problems Lessons from the Firm Primal and Dual again Consumer: Optimisation Reusing results on optimisation

A lesson from the firm z1z1 z2z2 z* ll ll q x1x1 x2x2 x* ll ll    Compare cost-minimisation for the firm...  ...and for the consumer   The difference is only in notation   So their solution functions and response functions must be the same Run through formal stuff

  U(x)  U(x) + [  – U(x)] Cost-minimisation: strictly quasiconcave U    U 1 (x  ) = p 1  U 2 (x  ) = p 2 … … …  U n (x  ) = p n  = U(x  ) one for each good n  p i x i i=1   Use the objective function  ...and output constraint  ...to build the Lagrangean l l Minimise   Differentiate w.r.t. x 1,..., x n and set equal to 0. l l Because of strict quasiconcavity we have an interior solution. utility constraint utility constraint  ... and w.r.t Lagrange multiplier      l l A set of n+1 First-Order Conditions   Denote cost minimising values with a *.

If ICs can touch the axes... n  p i x i i=1 + [  – U(x)] l l Minimise Can get “<” if optimal value of this good is 0 Interpretation l l A set of n+1 First-Order Conditions    U 1 (x  )  p 1  U 2 (x  )  p 2 … … …  U n (x  )  p n  = U(x  ) l l Now there is the possibility of corner solutions.

From the FOC U i (x  ) p i ——— = — U j (x  ) p j l l MRS = price ratio   “implicit” price = market price l l If both goods i and j are purchased and MRS is defined then... U i (x  ) p i ———  — U j (x  ) p j l l If good i could be zero then... MRS  price ratio   “implicit” price  market price Solution

The solution... l l Solving the FOC, you get a cost-minimising value for each good... x i * = H i (p,  ) * = *(p,  ) l l...for the Lagrange multiplier l l...and for the minimised value of cost itself. l l The consumer’s cost function or expenditure function is defined as C(p,  ) := min  p i x i vector of goods prices vector of goods prices Specified utility level {U(x)  }

The cost function has the same properties as for the firm Non-decreasing in every price. Increasing in at least one price Non-decreasing in every price. Increasing in at least one price Increasing in utility . Increasing in utility . Concave in p Concave in p Homogeneous of degree 1 in all prices p. Homogeneous of degree 1 in all prices p. Shephard's lemma. Shephard's lemma. Jump to “Firm”

Other results follow Shephard's Lemma gives demand as a function of prices and utility   H i (p,  ) = C i (p,  ) H is the “compensated” or conditional demand function. Properties of the solution function determine behaviour of response functions. Downward-sloping with respect to its own price, etc… “Short-run” results can be used to model side constraints For example rationing.

Comparing firm and consumer n min  p i x i x i=1 + [  – U(x)] Cost-minimisation by the firm... Cost-minimisation by the firm......and expenditure-minimisation by the consumer...and expenditure-minimisation by the consumer...are effectively identical problems....are effectively identical problems. So the solution and response functions are the same: So the solution and response functions are the same: x i * = H i (p,  ) C(p,  ) m min  w i z i z i=1 + [q –  (z)]   Solution function: C(w, q) z i * = H i (w, q)   Response function:   Problem: FirmConsumer

Overview... Primal and Dual problems Lessons from the Firm Primal and Dual again Consumer: Optimisation Exploiting the two approaches

n U(x) +  [ y –  p i x i ] i=1 n U(x) +  [ y –  p i x i ] i=1 The Primal and the Dual… l l There’s an attractive symmetry about the two approaches to the problem l l …constraint in the primal becomes objective in the dual… l l …and vice versa. l l In both cases the ps are given and you choose the xs. But… n  p i x i + [  – U(x)] i=1 n  p i x i + [  – U(x)] i=1

A neat connection x1x1 x2x2 x* ll ll    Compare the primal problem of the consumer...  ...with the dual problem   The two are equivalent   So we can link up their solution functions and response functions Run through the primal x1x1 x2x2 l l x*

U(x)U(x) Utility maximisation   U 1 (x  ) =  p 1 U 2 (x  ) =  p 2 … … … U n (x  ) =  p n one for each good n +  [ y –  p i x i ] i=1   Use the objective function  ...and budget constraint  ...to build the Lagrangean l l Maximise   Differentiate w.r.t. x 1,..., x n and set equal to 0. budget constraint budget constraint  ... and w.r.t  Lagrange multiplier      l l A set of n+1 First-Order Conditions   Denote cost minimising values with a *. n y   p i x i i=1 n y =  p i x i i=1 l l If U is strictly quasiconcave we have an interior solution. Interpretation If U not strictly quasiconcave then replace “=” by “  ”

From the FOC U i (x  ) p i ——— = — U j (x  ) p j l l MRS = price ratio   “implicit” price = market price l l If both goods i and j are purchased and MRS is defined then... U i (x  ) p i ———  — U j (x  ) p j l l If good i could be zero then... MRS  price ratio   “implicit” price  market price Solution ((same as before)

The solution... l l Solving the FOC, you get a cost-minimising value for each good... x i * = D i (p, y )  * =  *(p, y ) l l...for the Lagrange multiplier l l...and for the maximised value of utility itself. l l The indirect utility function is defined as V(p, y) := max U(x) vector of goods prices vector of goods prices money income {  p i x i  y}

A useful connection The indirect utility function maps prices and budget into maximal utility    = V(p, y) The indirect utility function works like an "inverse" to the cost function The cost function maps prices and utility into minimal budget   y = C(p,  ) The two solution functions have to be consistent with each other. Two sides of the same coin Therefore we have:  = V(p, C(p,  )) y = C(p, V(p, y)) Odd-looking identities like these can be useful

The Indirect Utility Function has some familiar properties... Non-increasing in every price. Decreasing in at least one price Non-increasing in every price. Decreasing in at least one price Increasing in income y. Increasing in income y. quasi-convex in prices p quasi-convex in prices p Homogeneous of degree zero in (p, y) Homogeneous of degree zero in (p, y) Roy's Identity Roy's Identity (All of these can be established using the known properties of the cost function) But what’s this…?

Roy's Identity  = V(p, y) = V(p, C(p,  )) 0 = V i (p,C(p,  )) + V y (p,C(p,  )) C i (p,  ) 0 = V i (p, y) + V y (p, y) x i *   Use the definition of the optimum   Differentiate w.r.t. p i. “function-of-a- function” rule   Use Shephard’s Lemma   Rearrange to get… V i (p, y) x i * = – ———— V y (p, y) V i (p, y) x i * = – ———— V y (p, y) Marginal disutility of price i Marginal utility of money income   So we also have… x i * = –V i (p, y)/V y (p, y) = D i (p, y) Ordinary demand function

Utility and expenditure n min  p i x i x i=1 + [  – U(x)] Utility maximisation Utility maximisation...and expenditure-minimisation by the consumer...and expenditure-minimisation by the consumer...are effectively identical problems....are effectively identical problems. So the solution and response functions are the same: So the solution and response functions are the same: x i * = H i (p,  ) C(p,  )   Solution function: V(p, y) x i * = D i (p, y)   Response function:   Problem: PrimalDual n max U(x) +  [ y –  p i x i ] x i=1

Summary A lot of the basic results of the consumer theory can be found without too much hard work. A lot of the basic results of the consumer theory can be found without too much hard work. We need two “tricks”: We need two “tricks”: 1. A simple relabelling exercise:  cost minimisation is reinterpreted from output targets to utility targets. 2. The primal-dual insight:  utility maximisation subject to budget is equivalent to cost minimisation subject to utility.

1. Cost minimisation: two applications n n THE FIRM n n min cost of inputs n n subject to output target Solution is of the form C(w,q) n n THE CONSUMER n n min budget n n subject to utility target Solution is of the form C(p,  )

2. Consumer: equivalent approaches n n PRIMAL n n max utility n n subject to budget constraint Solution is a function of (p,y) n n DUAL n n min budget n n subject to utility constraint Solution is a function of (p,  )

Basic functional relations C(p,  ) H i (p,  ) l l V(p, y) l l D i (p, y) cost (expenditure) Compensated demand for good i indirect utility ordinary demand for input i H is also known as "Hicksian" demand. Review Utility money income

What next? Examine the response of consumer demand to changes in prices and incomes. Examine the response of consumer demand to changes in prices and incomes.response of consumer demandresponse of consumer demand Household supply of goods to the market. Household supply of goods to the market. Develop the concept of consumer welfare Develop the concept of consumer welfareconsumer welfareconsumer welfare